Proposition

ℰ\mathcal{E} is Boolean iff the only dense subtopos of ℰ\mathcal{E} is ℰ\mathcal{E} itself.

Proof. Suppose ℰ\mathcal{E} is Boolean. ℰ¬¬=ℰ\mathcal{E}_{\neg\neg}=\mathcal{E} is the smallest dense subtopos (cf. double negation). Conservely, suppose ℰ\mathcal{E} is not Boolean then ℰ¬¬\mathcal{E}_{\neg\neg} is a second dense subtopos.

Proposition

Proposition

Let ℰ\mathcal{E} be a topos. Then automorphisms of Ω\Omega correspond bijectively to closed Boolean subtoposes. The group operation on Aut(Ω)Aut(\Omega) corresponds to symmetric difference of subtoposes.